2006-9.md

course	course_year	question_number	tags	title	year
Groups, Rings and Modules	IB	9	IB 2006 Groups, Rings and Modules	2.II.11E	2006

(i) Prove the first Sylow theorem, that a finite group of order $p^{n} r$ with $p$ prime and $p$ not dividing the integer $r$ has a subgroup of order $p^{n}$.

(ii) State the remaining Sylow theorems.

(iii) Show that if $p$ and $q$ are distinct primes then no group of order $p q$ is simple.